Shaping regularization in geophysical estimation problems |

bei-fmg2
Left: time-migrated image. Right: The
corresponding migration velocity from automatic picking.
Figure 8. |
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The task of this example is to convert the time migration velocity to the interval velocity. I use the simple approach of Dix inversion (Dix, 1955) formulated as a regularized inverse problem (Valenciano et al., 2004). In this case, the forward operator in equation 11 is a weighted time integration. There is a choice in choosing the shaping operator .

Figure 9 shows the result of inversion with shaping by triangle smoothing. While the interval velocity model yields a good prediction of the measured velocity, it may not appear geologically plausible because the velocity structure does not follow the structure of seismic reflectors as seen in the migrated image.

Following the ideas of steering filters (Clapp et al., 1998,2004) and plane-wave construction (Fomel and Guitton, 2006), I estimate local slopes in the migration image using the method of plane-wave destruction (Fomel, 2002) and define a triangle plane-wave shaping operator using the method of the previous section. The result of inversion, shown in Figures 10 and 11, makes the estimated interval velocity follow the geological structure and thus appear more plausible for direct interpretation. Similar results were obtained by Fomel and Guitton (2006) using model parameterization by plane-wave construction but at a higher computational cost. In the case of shaping regularization, about 25 efficient iterations were sufficient to converge to the machine precision accuracy.

bei-dix
Left: estimated interval velocity. Right:
predicted migration velocity. Shaping by triangle smoothing.
Figure 9. |
---|

bei-shp
Left: estimated interval
velocity. Right: predicted migration velocity. Shaping by triangle
local plane-wave smoothing creates a velocity model consistent with
the reflector structure.
Figure 10. |
---|

bei-shpw
Seismic image from
Figure 8 overlaid on top of the interval velocity
model estimated with triangle plane-wave shaping regularization.
Figure 11. |
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Shaping regularization in geophysical estimation problems |

2013-07-26